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Israel Journal of Mathematics

, Volume 223, Issue 1, pp 205–259 | Cite as

The Hölder property for the spectrum of translation flows in genus two

  • Alexander I. Bufetov
  • Boris Solomyak
Article

Abstract

The paper is devoted to generic translation flows corresponding to Abelian differentials with one zero of order two on flat surfaces of genus two. These flows are weakly mixing by the Avila–Forni theorem. Our main result gives first quantitative estimates on their spectrum, establishing the Hölder property for the spectral measures of Lipschitz functions. The proof proceeds via uniform estimates of twisted Birkhoff integrals in the symbolic framework of random Markov compacta and arguments of Diophantine nature in the spirit of Salem, Erdős and Kahane.

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References

  1. [1]
    J. S. Athreya, Quantitative recurrence and large deviations for Teichmuller geodesic flow, Geom. Dedicata 119 (2006), 121᾿40.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Avila and V. Delecroix, Weak mixing directions in non-arithmetic Veech surfaces, J. Amer. Math. Soc. 29 (2016), 1167᾿208.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. Avila and G. Forni, Weak mixing for interval exchange transformations and translation flows, Ann. of Math. (2) 165 (2007), 637᾿64.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Avila, S. Gouëzel and J.-C. Yoccoz, Exponential mixing for the Teichmüller flow, Publ. Math. Inst. Hautes Études Sci. (2006), 143᾿11.Google Scholar
  5. [5]
    A. Avila and M. Viana, Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture, Acta Math. 198 (2007), 1᾿6.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    L. Barreira and Y. Pesin, Nonuniform hyperbolicity, Encyclopedia of Mathematics and its Applications, Vol. 115, Cambridge University Press, Cambridge, 2007, Dynamics of systems with nonzero Lyapunov exponents.Google Scholar
  7. [7]
    V. Berthé, W. Steiner and J. M. Thuswaldner, Geometry, dynamics, and arithmetic of s-adic shifts, arXiv:1410.0331, preprint (2014).Google Scholar
  8. [8]
    A. I. Bufetov, Limit theorems for special flows over Vershik transformations, Uspekhi Mat. Nauk 68 (2013), 3᾿0.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. I. Bufetov and B. M. Gurevich, Existence and uniqueness of a measure with maximal entropy for the Teichmüller flow on the moduli space of abelian differentials, Mat. Sb. 202 (2011), 3᾿2.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. I. Bufetov, Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichm üller flow on the moduli space of abelian differentials, J. Amer.Math. Soc. 19 (2006), 579᾿23.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    A. I. Bufetov, Limit theorems for translation flows, Ann. of Math. (2) 179 (2014), 431᾿99.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. I. Bufetov and B. Solomyak, On the modulus of continuity for spectral measures in substitution dynamics, Adv. Math. 260 (2014), 84᾿29.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    P. Erdös, On a family of symmetric Bernoulli convolutions, Amer. J. Math. 61 (1939), 974᾿76.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    P. Erdös, On the smoothness properties of a family of Bernoulli convolutions, Amer. J. Math. 62 (1940), 180᾿86.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    K. Falconer, Techniques in fractal geometry, John Wiley & Sons, Ltd., Chichester, 1997.zbMATHGoogle Scholar
  16. [16]
    S. Ferenczi, Rank and symbolic complexity, Ergodic Theory Dynam. Systems 16 (1996), 663᾿82.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    N. P. Fogg, Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Mathematics, Vol. 1794, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.Google Scholar
  18. [18]
    G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. (2) 155 (2002), 1᾿03.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    G. Forni and C. Ulcigrai, Time-changes of horocycle flows, J. Mod. Dyn. 6 (2012), 251᾿73.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    H. Furstenberg, Stationary processes and prediction theory, Annals of Mathematics Studies, No. 44, Princeton University Press, Princeton, N.J., 1960.Google Scholar
  21. [21]
    M. Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), 773᾿22.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S. Ito, A construction of transversal flows for maximal Markov automorphisms, Tokyo J. Math. 1 (1978), 305᾿24.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J.-P. Kahane, Sur la distribution de certaines séries aléatoires, (1971), 119᾿22.zbMATHGoogle Scholar
  24. [24]
    A. Katok, Interval exchange transformations and some special flows are not mixing, Israel J. Math. 35 (1980), 301᾿10.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    H. Masur, Interval exchange transformations and measured foliations, Ann. ofMath. (2) 115 (1982), 169᾿00.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179᾿10.MathSciNetGoogle Scholar
  27. [27]
    R. Salem, A remarkable class of algebraic integers. Proof of a conjecture of Vijayaraghavan, Duke Math. J. 11 (1944), 103᾿08.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    R. Salem, Algebraic numbers and Fourier analysis, D. C. Heath and Co., Boston, Mass., 1963.zbMATHGoogle Scholar
  29. [29]
    P. Shmerkin, On the exceptional set for absolute continuity of Bernoulli convolutions, Geom. Funct. Anal. 24 (2014), 946᾿58.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    B. Solomyak, On the spectral theory of adic transformations, in Representation theory and dynamical systems, Adv. Soviet Math., Vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 217᾿30.MathSciNetzbMATHGoogle Scholar
  31. [31]
    P. Varjú, Absolute continuity of bernoulli convolutions for algebraic parameters, arXiv:1602.00261, preprint (2016).Google Scholar
  32. [32]
    W. A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), 201᾿42.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    W. A. Veech, The metric theory of interval exchange transformations. I. Generic spectral properties, Amer. J. Math. 106 (1984), 1331᾿359.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    W. A. Veech, Decoding Rauzy induction: Bufetov’s question, Mosc. Math. J. 10 (2010), 647᾿57, 663.MathSciNetzbMATHGoogle Scholar
  35. [35]
    A. M. Vershik, A theorem on Markov periodic approximation in ergodic theory, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 115 (1982), 72᾿2, 306, Boundary value problems of mathematical physics and related questions in the theory of functions, 14.MathSciNetzbMATHGoogle Scholar
  36. [36]
    A. M. Vershik, The adic realizations of the ergodic actions with the homeomorphisms of the Markov compact and the ordered Bratteli diagrams, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 223 (1995), 120᾿6, 338.zbMATHGoogle Scholar
  37. [37]
    A. M. Vershik and A. N. Livshits, Adic models of ergodic transformations, spectral theory, substitutions, and related topics, in Representation theory and dynamical systems, Adv. Soviet Math., Vol. 9, Amer. Math. Soc., Providence, RI, 1992, pp. 185᾿04.zbMATHGoogle Scholar
  38. [38]
    M. Viana, Interval exchange transformations and teichmüller flows, IMPA, preprint, 2008.Google Scholar
  39. [39]
    A. Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble) 46 (1996), 325᾿70.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  1. 1.Aix-Marseille UniversitéCNRS, Centrale Marseille, I2M, UMR 7373MarseilleFrance
  2. 2.Steklov Mathematical Institute of RASMoscowRussia
  3. 3.Institute for Information Transmission ProblemsMoscowRussia
  4. 4.National Research University Higher School of EconomicsMoscowRussia
  5. 5.Department of MathematicsBar-Ilan UniversityRamat-GanIsrael

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